# American Institute of Mathematical Sciences

October  2021, 20(10): 3445-3479. doi: 10.3934/cpaa.2021113

## Choquard equations via nonlinear rayleigh quotient for concave-convex nonlinearities

 1 Universidade Federal de Goiás, IME, Goiânia-GO, Brazil 2 Universidade Federal de Jataí, Jataí-GO, Brazil

* Corresponding author

Received  June 2020 Revised  May 2021 Published  October 2021 Early access  June 2021

Fund Project: The second author was partially supported by CNPq and FAPDF with grants 309026/2020-2 and 16809.78.45403.25042017, respectively

It is established existence of ground and bound state solutions for Choquard equation considering concave-convex nonlinearities in the following form
 $\begin{equation*} \begin{cases} -\Delta u +V(x) u = (I_\alpha* |u|^p)|u|^{p-2}u+ \lambda |u|^{q-2}u \, {\rm{\;in\;}}\, \mathbb{R}^N, \\ \ u\in H^1( \mathbb{R}^N) \end{cases} \end{equation*}$
where
 $\lambda > 0, N \geq 3, \alpha \in (0, N)$
. The potential
 $V$
is a continuous function and
 $I_\alpha$
denotes the standard Riesz potential. Assume also that
 $1 < q < 2$
,
 $2_\alpha < p < 2^*_\alpha$
where
 $2_\alpha = (N+\alpha)/N$
,
 $2_\alpha = (N+\alpha)/(N-2)$
. Our main contribution is to consider a specific condition on the parameter
 $\lambda > 0$
taking into account the nonlinear Rayleigh quotient. More precisely, there exists
 $\lambda^* > 0$
such that our main problem admits at least two positive solutions for each
 $\lambda \in (0, \lambda^*]$
. In order to do that we combine Nehari method with a fine analysis on the nonlinear Rayleigh quotient. The parameter
 $\lambda^*> 0$
is optimal in some sense which allow us to apply the Nehari method.
Citation: M. L. M. Carvalho, Edcarlos D. Silva, C. Goulart. Choquard equations via nonlinear rayleigh quotient for concave-convex nonlinearities. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3445-3479. doi: 10.3934/cpaa.2021113
##### References:
 [1] C. O. Alves and Ji anfu Yang, Existence and regularity of solutions for a Choquard equation with zero mass, Milan J. Math. Vol., 86 (2018), 329-342.  doi: 10.1007/s00032-018-0289-x. [2] A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.  doi: 10.1006/jfan.1994.1078. [3] M. Badiale and E. Serra, Semilinear Elliptic Equations for Beginners. Existence Results via the Variational Approach, Universitext. Springer, London, 2011. doi: 10.1007/978-0-85729-227-8. [4] T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbb{R}^{N}$, Commun. Partial Differ. Equ., 20 (1995), 1725–1741. doi: 10.1080/03605309508821149. [5] K. J. Brown and T. F. Wu, A fibering map approach to a semilinear elliptic boundary value problem, Electr. J. Differ. Equ., 69 (2007), 1-9. [6] K. J. Brown and T. F. Wu, A fibering map approach to a potential operator equation and its applications, Differ. Int. Equ., 22 (2009), 1097-1114. [7] M. L. M. Carvalho, Y. Ilyasov and C. A. Santos, Separating of critical points on the Nehari manifold via the nonlinear generalized Rayleigh quotients, arXiv: 1906.07759. [8] Yi-Hsin Cheng and Tsung-Fang Wu, Multiplicity and concentration of positive solutions for semilinear elliptic equations with steep potential, Commun. Pure Appl. Anal., 15 (2016), 1534-0392.  doi: 10.3934/cpaa.2016044. [9] S. Chen and X. Tang, Ground state solutions for general Choquard equations with a variable potential and a local nonlinearity, Rev. R. Acad. Cienc. Exactas Fìs. Nat. Ser. A Mat. RACSAM, 114 (2020), 14 pp. doi: 10.1007/s13398-019-00775-5. [10] P. Drábek and J. Milota, Methods of Nonlinear Analysis, Basler Lehrbücher, 2013. doi: 10.1007/978-3-0348-0387-8. [11] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2015. [12] E. P. Gross, Physics of Many-Particle Systems, Gordon Breach, New York, 1996. [13] Y. Huang, Tsung-Fang Wu and Y. Wu, Multiple positive solutions for a class of concave-convex elliptic problems in $\mathbb{R}^N$ involving sign-changing weight, II, Commun. Contemp. Math., 17 (2015), 1450045. doi: 10.1142/S021919971450045X. [14] X. Li, X. Liu and S. Mab, Infinitely many bound states for Choquard equations with local nonlinearities, Nonlinear Anal., 189 (2019), 111583. doi: 10.1016/j.na.2019.111583. [15] V. Moroz and J. Van Schaftingen, A guide to the Choquard equation, J. Fixed Point Theory Appl., 19 (2017), 773-813.  doi: 10.1007/s11784-016-0373-1. [16] V. Moroz and J. Van Schaftingen, Ground states of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184.  doi: 10.1016/j.jfa.2013.04.007. [17] Z. Nehari, On a class of nonlinear second-order differential equations, Trans. Amer. Math. Soc., 95 (1960), 101-123.  doi: 10.2307/1993333. [18] Z. Nehari, Characteristic values associated with a class of non-linear second-order differential equations, Acta Math., 105 (1961), 141-175.  doi: 10.1007/BF02559588. [19] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Stud. Appl. Math., 57 (1977), 93-105.  doi: 10.1002/sapm197757293. [20] X. Li and S. Ma, Choquard equations with critical nonlinearities, Communications in Contemporary Mathematics, 22 (2020), 1950023. doi: 10.1142/S0219199719500238. [21] R. Penrose, On gravity's role in quantum state reduction, Gen. Relativity Gravitation, 28 (1996), 581-600.  doi: 10.1007/BF02105068. [22] S. I. Pokhozhaev, The fibration method for solving nonlinear boundary value problems, Trudy Mat. Inst. Steklov., 192 (1990), 146-163. [23] P. Drabek and S. I. Pohozaev, Positive solutions for the p-Laplacian: application of the fibering method, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 703-726.  doi: 10.1017/S0308210500023787. [24] S. Pekar, Untersuchung Ber Die Elektronentheorie Der Kristalle, Akademie Verlag, Berlin, 1954. [25] P. Pucci and J. Serrin, The maximum principle, in Nonlinear Differential Equations and their Applications, Birkhäuser Verlag, Basel, 2007. [26] P. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, Conf. Board of Math. Sci. Reg. Conf. Ser. in Math., No. 65, Amer. Math. Soc., 1986. doi: 10.1090/cbms/065. [27] P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43, (1992), 270–291. doi: 10.1007/BF00946631. [28] C. A. Santos, R. L. Alves and K. Silva, Multiplicity of negative-energy solutions for singular-superlinear Schrödinger equations with indefinite-sign potential, (To appear in Communications in Contemporary Mathematics). [29] M. Struwe, Variational methods Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer Verlag, Berlin, 2000. doi: 10.1007/978-3-662-04194-9. [30] Y. Il'yasov and K. Silva, On branches of positive solutions for p-Laplacian problems at the extreme value of Nehari manifold method, Proc. Amer. Math. Soc., 146 (2018), 2925-2935.  doi: 10.1090/proc/13972. [31] Y. Il'yasov, On extreme values of Nehari manifold method via nonlinear Rayleigh's quotient, Topol. Methods Nonlinear Anal., 49 (2017), 683-714.  doi: 10.12775/tmna.2017.005. [32] Y. Il'yasov, On nonlocal existence results for elliptic equations with convex-concave nonlinearities, Nonl. Anal.: Th., Meth. Appl., 61 (2005), 211-236.  doi: 10.1016/j.na.2004.10.022. [33] M. Willem, Minimax Theorems, Birkhauser Boston, Basel, Berlin, 1996. doi: 10.1007/978-1-4612-4146-1. [34] Tsung-Fang Wu, Multiple positive solutions for a class of concave-convex elliptic problems in $\mathbb{R}^N$ involving sign-changing weight, J Funct. Anal., 258 (2010), 99-131.  doi: 10.1016/j.jfa.2009.08.005.

show all references

##### References:
 [1] C. O. Alves and Ji anfu Yang, Existence and regularity of solutions for a Choquard equation with zero mass, Milan J. Math. Vol., 86 (2018), 329-342.  doi: 10.1007/s00032-018-0289-x. [2] A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.  doi: 10.1006/jfan.1994.1078. [3] M. Badiale and E. Serra, Semilinear Elliptic Equations for Beginners. Existence Results via the Variational Approach, Universitext. Springer, London, 2011. doi: 10.1007/978-0-85729-227-8. [4] T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbb{R}^{N}$, Commun. Partial Differ. Equ., 20 (1995), 1725–1741. doi: 10.1080/03605309508821149. [5] K. J. Brown and T. F. Wu, A fibering map approach to a semilinear elliptic boundary value problem, Electr. J. Differ. Equ., 69 (2007), 1-9. [6] K. J. Brown and T. F. Wu, A fibering map approach to a potential operator equation and its applications, Differ. Int. Equ., 22 (2009), 1097-1114. [7] M. L. M. Carvalho, Y. Ilyasov and C. A. Santos, Separating of critical points on the Nehari manifold via the nonlinear generalized Rayleigh quotients, arXiv: 1906.07759. [8] Yi-Hsin Cheng and Tsung-Fang Wu, Multiplicity and concentration of positive solutions for semilinear elliptic equations with steep potential, Commun. Pure Appl. Anal., 15 (2016), 1534-0392.  doi: 10.3934/cpaa.2016044. [9] S. Chen and X. Tang, Ground state solutions for general Choquard equations with a variable potential and a local nonlinearity, Rev. R. Acad. Cienc. Exactas Fìs. Nat. Ser. A Mat. RACSAM, 114 (2020), 14 pp. doi: 10.1007/s13398-019-00775-5. [10] P. Drábek and J. Milota, Methods of Nonlinear Analysis, Basler Lehrbücher, 2013. doi: 10.1007/978-3-0348-0387-8. [11] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2015. [12] E. P. Gross, Physics of Many-Particle Systems, Gordon Breach, New York, 1996. [13] Y. Huang, Tsung-Fang Wu and Y. Wu, Multiple positive solutions for a class of concave-convex elliptic problems in $\mathbb{R}^N$ involving sign-changing weight, II, Commun. Contemp. Math., 17 (2015), 1450045. doi: 10.1142/S021919971450045X. [14] X. Li, X. Liu and S. Mab, Infinitely many bound states for Choquard equations with local nonlinearities, Nonlinear Anal., 189 (2019), 111583. doi: 10.1016/j.na.2019.111583. [15] V. Moroz and J. Van Schaftingen, A guide to the Choquard equation, J. Fixed Point Theory Appl., 19 (2017), 773-813.  doi: 10.1007/s11784-016-0373-1. [16] V. Moroz and J. Van Schaftingen, Ground states of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184.  doi: 10.1016/j.jfa.2013.04.007. [17] Z. Nehari, On a class of nonlinear second-order differential equations, Trans. Amer. Math. Soc., 95 (1960), 101-123.  doi: 10.2307/1993333. [18] Z. Nehari, Characteristic values associated with a class of non-linear second-order differential equations, Acta Math., 105 (1961), 141-175.  doi: 10.1007/BF02559588. [19] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Stud. Appl. Math., 57 (1977), 93-105.  doi: 10.1002/sapm197757293. [20] X. Li and S. Ma, Choquard equations with critical nonlinearities, Communications in Contemporary Mathematics, 22 (2020), 1950023. doi: 10.1142/S0219199719500238. [21] R. Penrose, On gravity's role in quantum state reduction, Gen. Relativity Gravitation, 28 (1996), 581-600.  doi: 10.1007/BF02105068. [22] S. I. Pokhozhaev, The fibration method for solving nonlinear boundary value problems, Trudy Mat. Inst. Steklov., 192 (1990), 146-163. [23] P. Drabek and S. I. Pohozaev, Positive solutions for the p-Laplacian: application of the fibering method, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 703-726.  doi: 10.1017/S0308210500023787. [24] S. Pekar, Untersuchung Ber Die Elektronentheorie Der Kristalle, Akademie Verlag, Berlin, 1954. [25] P. Pucci and J. Serrin, The maximum principle, in Nonlinear Differential Equations and their Applications, Birkhäuser Verlag, Basel, 2007. [26] P. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, Conf. Board of Math. Sci. Reg. Conf. Ser. in Math., No. 65, Amer. Math. Soc., 1986. doi: 10.1090/cbms/065. [27] P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43, (1992), 270–291. doi: 10.1007/BF00946631. [28] C. A. Santos, R. L. Alves and K. Silva, Multiplicity of negative-energy solutions for singular-superlinear Schrödinger equations with indefinite-sign potential, (To appear in Communications in Contemporary Mathematics). [29] M. Struwe, Variational methods Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer Verlag, Berlin, 2000. doi: 10.1007/978-3-662-04194-9. [30] Y. Il'yasov and K. Silva, On branches of positive solutions for p-Laplacian problems at the extreme value of Nehari manifold method, Proc. Amer. Math. Soc., 146 (2018), 2925-2935.  doi: 10.1090/proc/13972. [31] Y. Il'yasov, On extreme values of Nehari manifold method via nonlinear Rayleigh's quotient, Topol. Methods Nonlinear Anal., 49 (2017), 683-714.  doi: 10.12775/tmna.2017.005. [32] Y. Il'yasov, On nonlocal existence results for elliptic equations with convex-concave nonlinearities, Nonl. Anal.: Th., Meth. Appl., 61 (2005), 211-236.  doi: 10.1016/j.na.2004.10.022. [33] M. Willem, Minimax Theorems, Birkhauser Boston, Basel, Berlin, 1996. doi: 10.1007/978-1-4612-4146-1. [34] Tsung-Fang Wu, Multiple positive solutions for a class of concave-convex elliptic problems in $\mathbb{R}^N$ involving sign-changing weight, J Funct. Anal., 258 (2010), 99-131.  doi: 10.1016/j.jfa.2009.08.005.
$\lambda\in (0,\lambda_*)$
$\lambda = \lambda_*$
$\lambda\in(\lambda_*,\lambda^*)$
The functions $Q_n(t)$, $Q_e(t)$
$\lambda\in(0,\lambda_*)$
$\lambda = \lambda_*$
$\lambda\in(\lambda_*,\lambda^*)$
$\lambda_1<\lambda_2$
$\lambda_1<\lambda_2$
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